Synchronization of Kuramoto Oscillators via Cutset Projections

Synchronization in coupled oscillators networks is a remarkable phenomenon of relevance in numerous fields. For Kuramoto oscillators the loss of synchronization is determined by a trade-off between coupling strength and oscillator heterogeneity. Despite extensive prior work, the existing sufficient conditions for synchronization are either very conservative or heuristic and approximate. Using an oblique projection operator, called the cutset projection, we propose a novel family of sufficient synchronization conditions; these conditions rigorously identify the correct functional form of the trade-off between coupling strength and oscillator heterogeneity. To overcome the need to solve a nonconvex optimization problem, we then provide two explicit bounding methods, thereby obtaining (i) the best-known sufficient condition based on the 2-norm, and (ii) the first-known generally-applicable sufficient condition based on the ∞-norm. We conclude with a comparative study of the novel conditions for specific topologies and IEEE test cases; for most IEEE test cases our new sufficient conditions are one to two orders of magnitude more accurate than previous rigorous tests.